Optimal Consumption and Retirement Time under Shortfall Risk Measure

2026-06-17

AI Summary

This study investigates how investors jointly optimize investment, consumption, and endogenous early retirement decisions under relative performance evaluation while controlling the maximum wealth drawdown risk relative to a benchmark. By formulating a lifetime expected utility maximization problem, the authors transform the original problem into a mixed stochastic control and optimal stopping problem with a reflected state variable, which is then equivalently reduced—via convex duality—to a two-dimensional pure optimal stopping problem. Introducing an auxiliary reflected process, they innovatively characterize the geometric structure of the stopping region and derive in closed-form the optimal retirement boundary and associated feedback investment–consumption strategies. Key findings reveal that post-retirement investment becomes more conservative while consumption turns more aggressive; the maximum drawdown risk exhibits a U-shaped relationship with wealth; and drawdown costs, labor income, and leisure preferences significantly influence both retirement timing and strategic choices.

Abstract

This paper studies the optimal portfolio, consumption, and endogenous early retirement problem within a benchmark tracking framework by incorporating a new relative performance evaluation. In this framework, the investor maximizes expected lifetime consumption utility while managing the maximum wealth shortfall relative to a benchmark, with shortfall-management costs that may differ before and after retirement. Mathematically, the problem is a hybrid stochastic control problem involving both regular controls and an optimal stopping time, in which the running maximum process records the investor's largest benchmark shortfall. We introduce an auxiliary reflected state process and establish an equivalent hybrid stochastic control problem. By proving the convex duality theorem, we technically transform the original problem into a two-dimensional pure optimal stopping problem with state reflection. This enables us to characterize the geometric structure of the stopping set and derive the feedback-form optimal retirement boundary, as well as optimal portfolio and consumption policies. Analytical examples and numerical simulations reveal a two-stage structure with more conservative investment and more aggressive consumption after retirement. Driven by the retirement option, the expected largest shortfall risk follows a pronounced U-shaped pattern with respect to wealth. Shortfall management costs, labor income, and leisure preference significantly influence retirement timing, investment, and consumption.

Problem

• optimal retirement
• consumption
• portfolio choice
• shortfall risk
• benchmark tracking

Innovation

• hybrid stochastic control
• optimal stopping
• reflected state process
• convex duality
• benchmark tracking